What is Derivatives: Definition and 1000 Discussions

In finance, a derivative is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation, or getting access to otherwise hard-to-trade assets or markets.
Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over-the-counter (off-exchange) or on an exchange such as the Chicago Mercantile Exchange, while most insurance contracts have developed into a separate industry. In the United States, after the financial crisis of 2007–2009, there has been increased pressure to move derivatives to trade on exchanges.
Derivatives are one of the three main categories of financial instruments, the other two being equity (i.e., stocks or shares) and debt (i.e., bonds and mortgages). The oldest example of a derivative in history, attested to by Aristotle, is thought to be a contract transaction of olives, entered into by ancient Greek philosopher Thales, who made a profit in the exchange. Bucket shops, outlawed in 1936, are a more recent historical example.

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  1. R

    Derivatives and Linear transformations

    Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
  2. wirefree

    Is differentiation a possible approach?

    Question: I have a function of time. Its expression has a constant 'b' in it. I am asked to ascertain how changing 'b' affects the function. Specifically, I have velocity as a function of time which accounts for drag forces; 'b' is the drag coefficient. I am asked to ascertain how changing 'b'...
  3. H

    Proof of equality of mixed partial derivatives

    In the proof, mean value theorem is used (in the equal signs following A). Hence, the conditions for the theorem to be true would be as follows: 1. ##\varphi(y)## is continuous in the domain ##[b, b+h]## and differentiable in the domain ##(b, b+h),## and hence ##f(x,y)## is continuous in the...
  4. L

    The (asserted) equivalence of first partial derivatives

    In the solution to a differential-equation problem -- proof of the existence of an integrating factor -- the following statements are made regarding a general function "u(xy)" [that is, a function of two variable that depends exclusively on the single factor "x*y"]...
  5. R

    Inverse Laplace Transform with e^{a s}

    Homework Statement How can I take the Inverse Laplace Transform of $F(s) = \frac{d}{ds}\left(\frac{1-e^{5s}}{s}\right)$? I have tried going with inverse of the derivative and convolution (even tried evaluating the derivative and go from there) but although I can get to some results none of them...
  6. U

    Derivatives in 3D and Dirac Delta

    For a research project, I have to take multiple derivatives of a Yukawa potential, e.g. ## \partial_i \partial_j ( \frac{e^{-m r}}{r} ) ## or another example is ## \partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} ) ## I know that, at least in the first example above, there will be a...
  7. nuuskur

    Proof: Local extremum implies partial derivatives = 0

    Homework Statement Let f\colon\mathbb{R}^m\to\mathbb{R}. All partial derivatives of f are defined at point P_0\colon = (x_1, x_2, ... , x_m). If f has local extremum at P_0 prove that \frac{\partial f}{\partial x_j} (P_0) = 0, j\in \{1, 2, ..., m\} Homework Equations Fermat's theorem: Let...
  8. P

    Position function and its derivatives

    Okay so I'm currently in cal 3. I've also taken physics 1 and 2. When I first saw the position function differentiated into velocity and then to acceleration I was awestruck. Math is beautiful and divinely structured. That's what I thought. But as I've gone on and worked with it more and more...
  9. ElectricKitchen

    Fundamental Relationship Between Time and Space Derivatives

    Many physical laws involve relationships between time derivatives to space derivatives of one or more quantities. For example, thermal conduction relates the thermal energy time rate of change [dQ/dt] to temperature space rate of change [dT/dx]. In fluid flow, the Navier-Stokes Theorem relates...
  10. P

    Change of variables and discrete derivatives

    Hey I am trying to evaluate d/dx, d/dy and d/dz of a wavefunction defined on a grid. I have the wavefunction defined on equally spaced points along three axes a=x+y-z, b=x-y+z and c=-x+y+z. I can therefore construct the derivative matrices d/da, d/db and d/dc using finite differences but I...
  11. teroenza

    Four Tensor Derivatives -- EM Field Lagrangian Density

    Homework Statement Given the Lagrangian density \Lambda = -\frac{1}{c}j^lA_l - \frac{1}{16 \pi} F^{lm}F_{lm} and the Euler-Lagrange equation for it \frac{\partial }{\partial x^k}\left ( \frac{\partial \Lambda}{\partial A_{i,k}} \right )- \frac{\partial \Lambda}{\partial A_{i}} =0...
  12. binbagsss

    Algebra /derivatives/ chain rule/

    Homework Statement ##J=r^{2}\dot{\phi}## [1] ##\dot{r^{2}}=E^{2}-1-\frac{J^{2}}{r^{2}}+\frac{2MJ^{2}}{r^{3}}+\frac{2M}{r}##. [2] (the context is geodesic equation GR, but I'm pretty sure this is irrelevant). where ##u=r^{-1}## Question: From these two equations to derive...
  13. R

    Derivatives of the Lagrangian in curved space

    Follow along at http://star-www.st-and.ac.uk/~hz4/gr/GRlec4+5+6.pdf and go to PDF page 9 or page 44 of the "slides." I'm trying to see how to go from the first to the third line. If we write the free particle Lagrangian and use q^i-dot and q^j-dot as the velocities and metric g_ij, how is it we...
  14. powerof

    Symmetry in second order partial derivatives and chain rule

    When can I do the following where ##h_{i}## is a function of ##(x_{1},...,x_{n})##? \frac{\partial}{\partial x_{k}}\frac{\partial f(h_{1},...,h_{n})}{\partial h_{m}}\overset{?}{=}\frac{\partial}{\partial h_{m}}\frac{\partial f(h_{1},...,h_{n})}{\partial x_{m}}\overset{\underbrace{chain\...
  15. C

    Calculating Velocity of a Cannonball at a Given Height

    Homework Statement A Cannonball is shot upward from the ground into the air at t=0 sec. With initial velocity of 50m/s. Its height above the ground in metres is given by s(t)=50t-4.9t^2 . ----What is the velocity of the cannonball when it is 100m above the ground on the way up? "says the...
  16. M

    Equations of state -- Partial derivatives & Expansivity

    Homework Statement Show that the coefficient of volume expansion can be expressed as β= -1÷ρ (∂ρ÷∂T) keeping P (pressure) constant Where rho is the density T is Temperature Homework Equations 1/v =ρ β= 1/v (∂v÷∂T) keeping P (pressure ) constant The Attempt at a Solution I started with...
  17. kostoglotov

    Partial derivatives Q involving homogeneity of degree n

    Homework Statement Show that if f is homogeneous of degree n, then x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} = nf(x,y) Hint: use the Chain Rule to diff. f(tx,ty) wrt t. 2. The attempt at a solution I know that if f is homogeneous of degree n then t^nf(x,y) =...
  18. D

    Proving the reciprocal relation between partial derivatives

    If three variables x,y and z are related via some condition that can be expressed as $$F(x,y,z)=constant$$ then the partial derivatives of the functions are reciprocal, e.g. $$\frac{\partial x}{\partial y}=\frac{1}{\frac{\partial y}{\partial x}}$$ Is the correct way to prove this the following...
  19. D

    Tangent vectors as directional derivatives

    I have a few conceptual questions that I'd like to clear up if possible. The first is about directional derivatives in general. If one has a function f defined in some region and one wishes to know the rate of change of that function (i.e. its derivative) along a particular direction in that...
  20. RJLiberator

    Partial Derivatives and Polar Coordinates

    Homework Statement Write the chain rule for the following composition using a tree diagram: z =g(x,y) where x=x(r,theta) and y=y(r,theta). Write formulas for the partial derivatives dz/dr and dz/dtheta. Use them to answer: Find first partial derivatives of the function z=e^x+yx^2, in polar...
  21. I

    Partial Derivatives Homework: w(u,v)=f(u)+g(v)

    Homework Statement let w(u,v) = f(u) + g(v) u(x,t) = x - at v(x,t) = x + at show that: \frac{\partial ^{2}w}{\partial t^{2}} = a^{2}\frac{\partial ^{2}w}{\partial x^{2}} The Attempt at a Solution w(x-at, x+at) = f(x-at) + g(x+at) \frac{\partial }{\partial t}(\frac{\partial w}{\partial...
  22. Calpalned

    Partial derivatives of level curves

    Homework Statement Let ##C## be a level curve of ##f## parametrized by t, so that C is given by ## x=u(t) ## and ##y = v(t)## Let ##w(t) = g(f(u(t), v(t))) ## Find the value of ##\frac{dw}{dt}## Homework Equations Level curves Level sets Topographic maps The Attempt at a Solution Is it true...
  23. K

    MHB Derivatives of symmetric expressions

    So I was bored in math class and came up with this series of related questions, that I cannot answer: Is there a clean expression for $f'(x),$ where $$f(x)=\prod_{i=1}^{n}\dfrac{(x-i)}{(x+i)}?$$ What about for $f''(x)?$ Or for $$f(x)=\prod_{i=1}^{n}\dfrac{(x^2-i)}{(x^2+i)}?$$
  24. R

    Component functions and coordinates of linear transformation

    Let A(a, b, c) and A'(a′,b′,c′) be two distinct points in R3. Let f from [0 , 1] to R3 be defined by f(t) = (1 -t) A + t A'. Instead of calling the component functions of f ,(f1, f2, f3) let us simply write f = (x, y, z). Express x; y; z in terms of the coordinates of A and A, and t. I thought...
  25. Coffee_

    Chain rule when taking vector derivatives

    Consider a function of several variables ##T=T(x_{1},...,x_{3N})## Let's say I have N vectors of the form ##\vec{r_{1}}=(x_1,x_{2},x_{3})## and ##x_j=x_j(q_1,...,q_n)##. Awkward inex usage but the point is just that the each variable is contained in exactly 1 vector. Is it correct to in...
  26. R

    How to Prove Differentiability in R2 Using the Derivative of a Function?

    Let U={(x,y) in R2:x2+y2<4}, and let f(x,y)=√.(4−x2−y2) Prove that f is differentiable, and find its derivative. I do know how to prove it is differentiable at a specific point in R2, but I could not generalize it to prove it differentiable on R2. Any hint?
  27. Math Amateur

    MHB Proof of Darboux's Theorem (IVT for Derivatives)

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of the Intermediate Value Theorem (IVT) for Derivatives (Darboux's Theorem). Stoll's statement of the IVT for Derivatives and its proof read as follows: In the above proof, Stoll argues that...
  28. A

    Time derivatives of sin and cos phi

    Homework Statement By using chain rule of differentiation, show that: $$ \frac{\mathrm{d} sin\phi }{\mathrm{d} t} = \dot{\phi} cos\phi , \frac{\mathrm{d} cos\phi }{\mathrm{d} t} = -\dot{\phi} sin\phi , $$ Homework EquationsThe Attempt at a Solution I got this right for a homework problem...
  29. nmsurobert

    Partial derivatives and complex numbers

    Homework Statement show that the following functions are differentiable everywhere and then also find f'(z) and f''(z). (a) f(z) = iz + 2 so f(z) = ix -y +2 then u(x,y) = 2-y, v(x,y) = x Homework Equations z=x+iy z=u(x,y) +iv(x,y) Cauchy-Riemann conditions says is differentiable everywhere...
  30. E

    Understanding Partial Derivatives of x^2 + y^2 < 1

    Homework Statement x^2 + y^2 < 1 Find the partial derivatives of the function. Homework Equations x^2 + y^2 < 1 The Attempt at a Solution @f/@x = 2x = 0 @f/@y = 2y = 0 4. Their solution @f/@x = 2x = 0 @f/@y = 2y + 1 = 0 5. My Problem I don't see how / why they get 2y + 1 for the...
  31. BiGyElLoWhAt

    Calculating Functional Derivatives: Understanding Notation and Examples

    If I understand what's going on (quite possibly I don't), I think my book is using bad (confusing) notation. Homework Statement As written: "Calculate ##\frac{\delta H[f]}{\delta f(z)} \ \text{where} \ H=\int G(x,y)f(y)dy##" and ##\frac{\delta H[f]}{\delta f(z)}## is the functional derivative...
  32. M

    MHB Derivative of xf(x) at x=4: -2

    I am having trouble getting started with this question. Suppose that f(4)=7 and f′(4)=−2. Use the product rule to find the derivative of xf(x) when x=4. Thanks
  33. D

    Understanding Position Derivatives: Does Logic Follow?

    This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point A to point B , we know the object must have had some velocity (1st derivative of position) during that trip. It's also true that the object had to have accelerated to gain that...
  34. nmsurobert

    Partial derivatives transformation

    Homework Statement Homework Equations included in the first picture The Attempt at a Solution i feel confident in my answer to part "a". i pretty much just did what the u and v example at the top of the page did. but for part "b" i tried to distribute and collect like terms and what not...
  35. D

    A question on proving the chain rule

    I'm currently reviewing my knowledge of calculus and trying to include rigourous (ish) proofs in my personal notes as I don't like accepting things in maths on face value. I've constructed a proof for the chain rule and was wondering if people wouldn't mind checking it and letting me know if it...
  36. dumbboy340

    Deep understanding of derivatives

    Hello everyone,i want to know about derivatives in detail!suppose a function say a=x^2 has derivative 2x,i want to know what does that mean?how we'll prove it?if we put x=2,then a=4 and if we put x=3,we'll get a=9,does that mean a=9-4=5,the change? Sorry for the long question.. Thanks!
  37. P

    Derivatives with respect to a Supernumber?

    So I've been trying to think about some papers in Supersymmetry and I need to somehow define a derivative of a supernumber, with respect to another supernumber. I mean a supernumber to be a number with an ordinary "body" and a "soul" which is a product of an even number of Grassmann numbers...
  38. Z

    Lie Dragging and Lie Derivatives

    Dear friends, I am currently studying some concepts on Differential Geometry using the book "Geometrical Methods of Mathematical Physics" by Bernard F. Schutz and have so far read up to the beginning of Chapter 3 entitled "Lie Derivatives and Lie Groups". Even though Chapters 1 and 2 are very...
  39. MidgetDwarf

    Help understanding the Chain Rule book for derivatives

    After completing calculus 2 with an A I now realize I know nothing of mathematics. We used stewart calculus and I did not really like it, due to a lot of hand waiving. I got an older edition of thomas calculus with analytic geometry 3rd ed, and so far I'm having a blast learning proofs from...
  40. Einj

    Riemann tensor and derivatives of ##g_{\mu\nu}##

    Hello everyone, I'm studying Weinberg's 'Gravitation and Cosmology'. In particular, in the 'Curvature' chapter it says that the Riemann tensor cannot depend on ##g_{\mu\nu}## and its first derivatives only since: What I don't understand is how introducing the second derivatives should change...
  41. gracy

    Explain Derivatives in Physics: Gauss Law

    Can anyone explain what actually use of derivatives in physics.It's totally beyond my understanding.I was doing gauss law and i came across this derivative doubt.In the video at time 8:13 to 8:33 what he means by saying if area is small electric field should be approximately constant?is he...
  42. M

    Differentiation of a sphere -- raindrop evaporating as it falls

    < Moderator Note -- Thread moved from the technical PF Calculus forum > I can't seem to grasp the idea of this problem, any help is much needed. The problem reads, "As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its...
  43. T

    Partial Derivatives: Solving y^2=uy-v

    Hey, Little confused by something: if we have u=x+y and v=xy what is the partial derivative w.r.t. u of y^2=uy-v I am told it is 2y (dy/du) = u (dy/du) + y And I can see where these terms come from. What I don't understand is why there is no (dv/du) term, as v and u aren't independent...
  44. K

    MHB Elementary Calculus Problem involving Derivatives

    I am currently working on old tests to prepare for my final in Elementary Calculus. I came across this problem and have no idea what to do. Any help would be greatly appreciated. A new Japanese restaurant is pricing a koi pond. A 4 foot deep, 8 foot radius circular pond looks nice, but...
  45. K

    Classifying Critical Points: Finding Local Extrema and Saddle Points

    Homework Statement Homework EquationsThe Attempt at a Solution 1) I found the asymptote as (+/- 1) 2) Let f(x) = y; dy/dx = -2x^2 / (x^4 - 2x^2 + 1) = 0 -2x^2 - 0 x = 0; Since f() != 1, f(2) > 0 Increasing Since f() != -1, f(-2) < 0 Decreasing So i guess range is increasing or x >=2...
  46. F

    Optimisation Problem Using Derivatives

    Homework Statement a cylindrical tin can with volume 0.3l is being made, with the top and bottom sufaces twice the thickness as the sides. Show that a height to radius ration of h=4r will minimise the amount of aluminium required. Homework Equations V=\pi r^2 h \\ A = 2 \pi r^2 + 2 \pi r h...
  47. S

    Homogeneity and derivatives

    I've been reading a book on economics and they defined a homogeneous function as: ƒ(x1,x2,…,xn) such that ƒ(tx1,tx2,…,txn)=tkƒ(x1,x2,…,xn) ..totally understandable.. they further explained that a direct result from this is that the partial derivative of such a function will be homogeneous to the...
  48. F

    Prove limit x approaches 0 of a rational function = ratio of derivatives

    1. The problem statement, all variables and given/known dat If f and g are differentiable functions with f(O) = g(0) = 0 and g'(O) not equal 0, show that lim f(x) = f'(0) x->0 g(x) g'(0) The Attempt at a Solution I know that lim as x→a f(a) = f(a) if function is continuous. since its...
  49. P

    Partial Derivatives multivariable

    I am quite new to the topic of multivariable calculus. I came across the concept of "gradient" (∇), and although the treatment was somewhat slapdash, I think I got the hang of it. Consider the following case: ##z = f(x,y,t)## ##∇z = \frac{∂z}{∂t} + \frac{∂z}{∂y} + \frac{∂z}{∂x}## If we're...
  50. D

    Difference between these 4-vector derivatives?

    Hey everyone, So I've come across something in my notes where it says that these two Lagrangian densities are equal: \mathcal{L}_{1}=(\partial_{mu}\phi)^{\dagger}(\partial^{\mu}\phi)-m^{2}\phi^{\dagger}\phi \mathcal{L}_{2}=-\phi^{\dagger}\Box\phi - m^{2}\phi^{\dagger}\phi where \Box =...
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